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A lily pad sits on a pond. It doubles in size every day. It takes 30
days for it to cover the pond. If you start with 8 lily pads instead,
how many days does it take to cover the pond?

I think that the answer is $27$, but I don't really think that makes sense intuitively. I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.

Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.

Hint $#1$:

At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.

In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?

(I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)

Hint $#2$:

If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.

Your answer is correct

If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$and$$2^{26}cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days.

The intuition is that:

If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.

$27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.

Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.

I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.

Lily pad doubles in size every day, so it is increasing as a geometric progression.
$$begin{array}{c|c|c|c|c|c|c|c|c}
text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
hline
text{size of $1$ lily pad}&1&2&2^2&2^3&cdots&2^{25}&2^{26}&2^{27}&2^{28}&color{red}{2^{29}} end{array}$$

If you start with $8$ lily pads, each doubling on its own, then:
$$begin{array}{c|c|c|c|c|c|c|c|c}
text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
hline
text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&cdots&2^{28}&color{red}{2^{29}}&2^{30}&2^{31}&2^{32} end{array}$$
Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.

As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.